Definitions
We let m be a measure on the σ-algebra of Borel sets of a Hausdorff topological space X.
The measure m is called inner regular or tight if m(B) is the supremum of m(K) for K a compact set contained in the Borel set B.
The measure m is called outer regular if m(B) is the infimum of m(U) for U an open set containing the Borel set B.
The measure m is called locally finite if every point has a neighborhood of finite measure.
The measure m is called a Radon measure if it is inner regular and locally finite.
(It is possible to extend the theory of Radon measures to non-Hausdorff spaces, essentially by replacing the word "compact" by "closed compact" everywhere. However, there seem to be almost no applications of this extension.)
Read more about this topic: Radon Measure
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